Pre-twisted algebrizable differential equations

TL;DR

This paper introduces a new concept of algebraic differentiability for functions into algebraic structures, develops related complex analysis tools, and applies these to solve certain partial differential equations.

Contribution

It defines algebraic algebraic differentiability, establishes generalized Cauchy-Riemann equations, and demonstrates applications to vector fields and PDE solutions.

Findings

algebraic algebraic differentiability is applicable to vector fields in billiards.

The framework enables constructing exact solutions to the 3D heat equation.

Generalized Cauchy-Riemann equations are derived for these functions.

Abstract

We introduce the \emph{$\varphi\mathbb{A}$-differentiability} for functions $f:U\subset \mathbb R^{k}\to\mathbb A$ where $\mathbb A$ is the linear space $\mathbb R^{n}$ endowed with an algebra product which is unital, associative, commutative, $U$ is an open set, and $\varphi:U\subset \mathbb R^{k}\to\mathbb A$ is a differentiable function in the usual sense. We also introduce the corresponding generalized Cauchy-Riemann equations ($\varphi\mathbb{A}$-CREs), the Cauchy-integral theorem, and the \emph{$\varphi\mathbb A$-differential equations}. The four-dimensional vector fields associated with triangular billiards are $\varphi\mathbb{A}$-differentiable. The \emph{$\varphi\mathbb A$-differential equations} can be used for constructing exact solutions of partial differential equations like the three-dimensional heat equation.